标题： 关于用偏微分方程系统逼近空间卷积的研究 显示英文标题

标题： On the approximation of spatial convolutions by PDE systems

摘要： 本文研究了具有给定径向积分核的空间卷积的近似问题。 先前的研究表明，利用偏微分方程（PDE）系统来近似空间卷积可以消除一维空间中积分公式带来的分析困难。 在本文中，我们建立了用于高维空间卷积的PDE系统近似方法。 对于任意给定的径向积分核，我们推导出适当的近似函数作为Green函数的线性组合。 为了验证该方法的有效性，我们引入适当的积分变换以证明由Green函数构造的基的完备性。 此框架能够使用PDE解的线性组合来近似具有任意径向积分核的非局部卷积型算子。 最后，我们给出了数值例子，展示了所提出方法的有效性。 显示英文摘要

摘要： This paper considers the approximation of spatial convolution with a given radial integral kernel. Previous studies have demonstrated that approximating spatial convolution using a system of partial differential equations (PDEs) can eliminate the analytical difficulties arising from integral formulations in one-dimensional space. In this paper, we establish a PDE system approximation for spatial convolutions in higher spatial dimensions. We derive an appropriate approximation function for given arbitrary radial integral kernels as a linear sum of Green functions. In establishing the validity of this methodology, we introduce an appropriate integral transformation to show the completeness of the basis constructed by the Green functions. This framework enables the approximation of nonlocal convolution-type operators with arbitrary radial integral kernels using linear sums of PDE solutions. Finally, we present numerical examples that illustrate the effectiveness of our proposed method.